CHAPTER 4
EVERYTHING YOU ALWAYS WANTED TO KNOW (AND MAYBE DIDN'T)
ODDS PERMUTATIONS, AND RETURN ON INVESTMENTS
WHAT ARE CARD ODDS?
If you haven’t already notice, probability is a huge factor in Texas Hold’em. For example, there are 2,598,960 possible hands in a 52 card deck but only 4 Royal Flushes. If the average serious poker player is dealt 100,000 hands in their lifetime, they will never hold (on the first five cards) more than 4 percent of all the possible hands. And likely a lot less.
Figuring out straight card combinations for the purpose of this text are called Card Odds (you will be introduced to other kinds of odds later). Card Odds can reveal some quite interesting information.
For example, how many pat straight flushes will you see in your lifetime? To determine that number, the expected number of hands that could be dealt during your lifetime is estimated by the following calculation.
10 complete poker hands / hr. x 5 hrs. / game x 50 games / yr. x 40 yrs. / poker life = 100,000 hands of poker per lifetime.
This is pretty aggressive estimate, as most people will never come close to this number of complete hands in Texas Hold’em. Based on this level of play, the number of pat (on the first five cards) poker hands that you should get during your lifetime is calculated from the card odds and tabulated as follows:
| Cards Dealt | Number of Pat Hands |
| No pair | 50,000 |
| One pair | 40,000 |
| Two pair | 5,000 |
| Three of a kind | 2,000 |
| Straight | 400 |
| Flush | 200 |
| Full house | 170 |
| Four of a kind | 25 |
| Straight flush | 1.4 |
| Royal straight flush | 0.15 |
So statistically, you should see a pat straight flush on your first five cards once or twice during your lifetime. Most average poker players will never see even one.
Players use card odds to make playing decisions. A decision made without taking into account card odds makes poker a guessing game. The chances of finishing a slush or a straight, the probability of getting an over card (face card), the percentage of times you’re going to flop a card to match your pocket pair are all extremely important factors in Texas Hold’em.
Knowledge of these statistics is key to winning.
Here are some other basic probabilities that you should know about:
You need one more heart to make your flush on the turn or river: 35%
Probability of hitting an open ended straight draw (i.e. 4 straight cards, need one on either end to hit on turn or river): 31.5%
Probability of being dealt suited cards: 23.5%
Probability of hitting a three or four of a kind at the flop when you hold a pocket pair: 11.8%
Probability you will make a pair at the flop, holding two unpaired cards in the hole: 32.4%
Probability of being dealt AA: .45%
Probability of no one holding a specific card, by number of players, assuming you do not have that card, by number of total players:
2- 84.5%
3- 70.9%
4- 59.0%
5- 48.6%
6- 39.7%
7- 32.1%
8- 25.6%
9- 20.1%
10- 15.6%
Probability someone else does not have an ace, assuming you do have an ace, by total number of players:
2- 88.2%
3- 77.5%
4- 67.6%
5- 58.6%
6- 50.4%
7- 43.0%
8- 36.4%
9- 30.5%
10- 25.3%
HOW ARE THE ODDS CALCULATED
Let’s look at the example of having 4 outs (four cards you need to make your hand). Say you’re holding 6c 7d and the flop come 9s 10h Kc. In this case you need an 8 to make the straight. Since there are four 8’s in the deck, you have 4 outs.
YOUR POCKET
THE FLOP
ODDS WITH ONE CARD TO COME
To calculate the appropriate odds with two cards to come, you must first determine the total number of two card combinations possible after the flop.
The easiest way to calculate this is by multiplying the number of cards available for the turn (47) by the number of cards available for the river (46) and dividing that number by 2 (because a card can’t match itself). 47*46/2 = 1081
A certain number of these 1081 two card combinations with have eights in them. To determine odds properly, you need to calculate two more figures.
EIGHTS ON BOTH THE TURN AND THE RIVER
One of the four eights can appear on the turn. And if one does, there will be three left for the river. If you multiply 4 by 3 and divide by 2 (because a card can’t match itself) you see that there are six unique pairing of 8s.
EIGHTS ON THE TURN OR RIVER
If an eight comes on the turn, there are 46 unseen cards remaining. But you’re no longer interested in the three remaining eights, so you can subtract those. This leaves 43 unseen cards that will make a unique pair with one of the eights. Multiply 4 (the number of 8s in the deck) by 43 (the number of unseen cards to arrive at 172.
FINISH THE CALCULATION
172 plus 6 comes to 178 the total number of two card combinations that have at least one eight in them and as many as two eights.
Out of 1081 possible two card combinations on the turn and river, 178 of those combinations help us make our hand. Subtract 178 from 1081 to find the number of combinations that don’t make the straight (1081 – 178 = 903).
The odds against making a straight by the river are: 903:178, or 20%.
What About The Cards The Other Players Are Holding?
Ever wonder why we never factor the opponents’ cards or the burn cards when figuring out how many cards are left?
The reason is that we only consider unseen cards. If you saw what the burn cards were, or an opponent showed you his hand, you would know that those cards are not going to be drawn and could use that. We typically do not know what they have, so we don’t even think about it when talking about odds.
For instance, take a standard deck of 52 cards, remove 2 Aces and burn 25 of them. If you drew the next card, what are the chances of it being an Ace? It would be 2/50 (2 Aces left out of 50 unseen cards). It would NOT be 2/25 just because you burned half the deck. Okay, do the same thing again, but this time you get to look at the burn cards. Let’s say that of all the cards you burned, none were an Ace. Now your odds are 2/25 because there are still 2 Aces and now only 25 unseen cards.
You will find that you can easily remember a few of the most common situations for outs such as the four flush or straight draw but there has to be an easier way than memorizing the figures for every number of outs. The good news is that there is a way to get a good estimation of the odds without the heavy math and you can also use handy odds charts.
WHAT HANDS WILL WIN THE POT?
The following are the most valuable starting hands in Texas Hold’em. This chart assumes a medium to loose $5 10 Texas Hold’em game. The results are based on a computer simulation of 5,000,000 played hands. The percentage shown indicates how many times in a typical game these hands win the pot.
| Pair | 31% |
| 2 Pair | 27% |
| Three of a kind | 12% |
| Straight | 9% |
| Flush | 9% |
| Full house | 9% |
| Bust (nothing) | 2% |
| Four of a kind | 1% |
| Straight flush | <1% |
| Royal flush | <1% |
WHAT ARE MY CHANCES OF WINNING?
What are my chances of winning a hand with a given set of pocket cards? What are the odds of filling a straight?
How much you can expect to profit from these starting hands?
The starting hand average win for each time played based on medium to loose Texas Hold’em Poker games with a $5/$10 limit:
AA---->$34.19
KK---->$24.13
QQ---->$17.36
JJ---->$12.08
AK suited---->$11.63
AK offsuit---->$8.65
AQ suited---->$8.32
1010--->$7.72
AJ suited---->$5.69
AQ offsuit---->$5.47
POCKET CARD ODDS
WHAT ARE POT ODDS?
Pot odds are the odds you get when analyzing the current size of the pot vs. you next call or bet. Pot Odds help you make important call, raise or fold decisions.
Example 1: There is $200 in the pot and a final $10 bet coming at you. You are looking to fill in your 4 card flush. Based on needing one of four suits, the short cut math is 1:4 chances or 25%.
Winning consistently is about beating Pot Odds and not over betting. If you Card Odds are 25% (you have four hearts and need another to win) and you only need to bet 5% of the present pot ($10 as a percentage of the $200) to see the last card, you are in great shape. Based on Pot Odds for this hand you could go as high as 25% of the pot based on your odds to pull a heart on the river.
YOUR POCKET
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Example 2: You are in a $5/$10 Hold’em game with J10 pocket cards and one opponent left on the turn.
THE FLOP
You have an outside straight draw with a board of 2 5 9 Q, and only the river card left to make you straight. Any 8 or any King will finish this straight for you, so you have 8 outs (four 8’s and 4 K’s left in the deck) and 46 unseen cards left. 8/46 gives you a 6:1 chance or a 17% chance of getting the win. Your opponent bets $10.
6/1 = 17% Pot Odds
If you take a $10 bet you could win $100. Your bet is then 10% of the final pot. The 10% bet is smaller than the 17% odds, so you are in good shape from an investment point of view.
Whenever your call or bet (in this case 10% of the pot) is smaller than your odds of getting the winning hand (17%) you are buying the next card at a discount. The key is looking at the odds of a winning hand, not building a part of a winning hand.
Too many beginners just play good cards and just toss bad ones, or if they have the prospect of a straight or a flush, they will just draw and draw with little consideration to anything else.
Unless you have the NUTS (nuts = the very best possible hand) every time there is a certain degree of risk when you put your money in the pot.
In the heat of the battle you may not always have time to get out your calculator and do the number crunching. You can look at Pot Odds in a much simpler way. Just ask yourself: “Is there enough money in the pot to justify the risk?”
If there is a big pot, unless you just know you are beat, or it simply costs too much to call (which means your pot odds may not be so great after it’s all said and done) try to get some action. Even if you knew you were a long shot, would you bet $25 or $50 on a chance to win $500 or $1000 or more? While you may be a long shot, it’s not a bad bet if there is a reasonable chance you could win.
On the flip side, and this is important, you may have a decent hand, and it may even be the winner, (use your opponent or ‘tells’ knowledge) but if it’s going to cost you $250 to call and you stand to only win a few hundred then your pot odds are not so great and it is probably not worth the risk.
Pot Odds ratios are a useful tool to see how often you need to win the hand to break even.
If there is $100 in the pot and it takes $10 to call, you must win this hand 1 out of 11 times in order to break even. The thinking goes along the lines of: If you play 11 times, it’ll cost you $110, but when you win, you get $110 ($100 + your $10 call).
The usefulness of card odds and pot odds becomes very apparent when you start comparing the two. As we know, in a flush draw, your card odds for making your flush are 1.9 to 1 or 35%. Let’s say you’re in a hand with a nut flush draw and it’s $5 to you on the flop to call. Do you call? Your answer should be: What are my pot odds?
If there is $15 in the pot plus a $5 bet from an opponent, then you are getting 20:5 or 4:1 pot odds – 25%. This means that in order to break even, you must win 1 out of every 5 times. However, with your flush draw, your odds of winning are 1 out of every 3 times! You should quickly realize that not only are you breaking even, but you’re making a nice profit on this too. Let’s calculate the profit margin on this by theoretically playing this hand 100 times from the flop, when is then checked to the river.
Total Cost to Play = 100 hands * $5 to call = $500
Pot Value = $15 + $5 call = $25
Odds to Win = 1.9:1 or 35% (from the flop)
Total Hands Won = 100 * Odds to Win (35%) = 35 wins
Net Profit = Net Cost to Play + (Total Times Won * Pot Value)
= $500 + (35 * $30)
= $500 + $1,050
= $550 Profit
As you can see, you have a great reason to play this flush draw, because you’ll be making money in the long run according to your card odds and pot odds. The most fundamental point to take from this is:
If your Pot Odds > Card Odds, then you are making a profit.
So, even though you may be faced with a gut shot straight draw at times, which is a terrible draw at 5 to 1 hand odds, it can be worth it to call if you are getting pot odds greater than 5 to 1. Other times, if you have an excellent draw such as the flush draw, but someone has just raised a large amount so your pot odds are 1:1 for instance, then you obviously should not continue trying to draw to a flush, as you will lose money in the long run. In this situation, a fold or semi-bluff is your only solution, unless you know there will be callers behind you that improve your pot odds to better than break even.
Your ability to memorize or calculate your card odds and figure out your pot odds will lead you to make many of the right decisions in the future. Just be sure to remember that fundamental principle of playing drawing hands when your pot odds are greater than your card odds.
Another good rule to follow: A lot of players want to somehow factor in money they wagered on previous rounds. With the last example, you probably had already invested a significant portion of the $200 pot. Let’s say $50. Does that mean you should play or fold because of that money you already have in there? $50/$200?
That’s a big no.
That’s not your money anymore! It’s in a pool of money to be given to the winner. You have no stake in that pot. The only stake you might have is totally mental and has no bearing on hard statistics.
The next step is to use bet odds and implied odds. That’s tougher, because it involves predicting reactions of other players. With bet odds, you try to factor in how many people are going to call a raise. With implied odds, you’re thinking about reactions for the rest of the game. One example on implied odds…Say it’s another $5/$10 Hold’em game and you have a four flush on the flop.
Your neighbor bets, and everyone folds. The pot is $50 at this point. First you figure out your chance of hitting your flush on the turn, and it comes out to about 19% (about 1 in 5). You have to call this $5 bet vs. a $50 pot, so that’s a 10x payout. 1/5 is higher than 1/10, so bet odds are okay, but you must consider that this guy’s going to bet into you on the turn and river also. That’s the $5 plus two more $10 bets.
So now you are facing $25 more till the end of the hand. So you have to consider your chances of hitting that flush on the turn or river, which makes it about 35% (better than 1 in 3 now), but you have to invest $25 for a finishing pot of $100. $100/$25 is 1 in 4. That’s pretty close.
But there’s more!
If you don’t make it on the turn, it’ll change you outs and odds. You’ll have a 19.6% chance of hitting the flush (little worse than 1 in 5), but a $20 investment for a finishing pot of $100! $100/$20 is 1 in 5. So the chances could take a nasty turn if you didn’t hit it! What makes it more complicated is that if you did hit it on the turn, you could raise him back, and get an extra $20 or maybe even $40 in the pot.
Once you’ve mastered simple outs and pot odds, bet and implied odds are just a longer extension of these equations. If you think about these things while you play, they will eventually become second nature to you.
More Odds Examples: A Pocket Pair
You start with a pair of Jacks in the pocket. Not too shabby. The flop however doesn’t contain another Jack.
YOUR POCKET
THE FLOP
Lesson 1: What’s my chance of getting a Jack on the turn?
You need to just figure out the number of outs and divide it by the number of cards in the deck. There are 2 more Jacks. There’s 47 more cards since you’ve seen five already. The answer is 2/47, or .0426, close to 4.3%.
Lesson 2: No luck on the turn, how about the river card?
Still 2 Jacks left, but one less card in the deck bringing the grand total to 46. What’s 2/46? That’s .0434, which is also close to 4.3%. Your chances didn’t change much.
Lesson 3: Forget just getting just one Jack! I want them both! What are my chances? Since we’re trying to figure out the chances of getting one on the turn AND the river, and not getting one on EITHER the turn or river, we don’t have to reverse out thinking. Just multiply the probability of each event happening. Chances of getting that first Jack on the turn was .0426, remember? The chance of getting a second Jack on the river would be 1/46, because there’ll only be one Jack left in the deck. That’s about .0217, or 2.2%. To get the answer, multiply them. .0426 X .0217 is about .0009! That’s around one tenth of a percent. I wouldn’t bank on that one.
Lesson 4: Hey, what were my chances of getting a pair of Jacks anyway? To figure that out, think of it as getting dealt one card, then another. What are your chances of the second card matching the first one? There will be 3 cards left like the one you have. There’s 51 cards left in the deck. 3/51 is .059 or 5.9%. What is the chance that 55 it’ll be Jacks? Well, there are 13 different cards. So, .059/13 I about .0045, a little less than half a percent.
Lesson 5: What were my chances of getting a Jack on the flop? Now you do have to think in reverse as in the previous example. Figure out the chances of NOT getting a Jack on each successive card flip. First card you have a 48/50 chance (48 non-Jack cards left, 50 cards left in the deck), second card is 47/49, third card is 46/48. Those come out to .96, .959, and .958. Multiply then and get .882, or an 88.2 % chance of NOT getting any Jacks on the flop. Invert it to figure out what your chances really are and you get .118 or 11.8%. This will be your chance to get one or two Jacks.
Example #2: The straight draw
THE POCKET
You start with a Jack of Spades and a Ten of Spades. You get a rainbow flop with a Queen of Spades, a Three of Diamonds, and a Nine of Clubs. You’ve got a straight draw.
THE FLOP
Lesson 1: What are my chances of hitting it on the next card? Same as before, but with different outs. A King or an Eight will complete your hand. There is presumably four of each left in the deck. You’ve got 8 outs. The chance of getting one of them on the turn is 8 over 47, because there’s 47 cards left in the deck. That comes out to about .170 or around 17%.
Lesson 2: I didn’t get it on the turn! What are my chances now? There are still 8 cards left in the deck that’ll help you, but 46 cards left in the deck. That’s 8 over 46. It changes to .174. It’s improved to a whopping 17.4%!
Lesson 3: I should have thought about my total chances first, I’m such an idiot. What are my chances of getting that card on the turn OR the river?
Once again we’ll have to calculate the chances of a King or Eight NOT appearing, so we can do it like the last problem (in this case {39/47} X {38/46}). Or, since we’ve already figured out our chances in the previous two lessons, we can just invert the probabilities and multiply them. You had a .170 chance on the turn, and a .174 on the river. By inverting, I mean subtracting them from one. Now we’ve got .830 and .826! Multiply and get .686! That’s our chance of NOT hitting our card at all. So invert it again and get .314, or 31.4%.
Example #3: Top two pair
THE POCKET
THE FLOP
You get dealt a King of Diamonds and a Nine of Hearts. The flop is looking pretty good for you with a King of Spades, a Nine of Clubs and a Four of Clubs. Top two pair!
Lesson 1: What are my chances of getting a full house on the turn?
To get a full house, you need another King or Nine to pop up. There are two of each left in the deck. So you’ve got 4 outs. After the flop there’s always 47 cards unaccounted for. 4/47 is around .085 or an 8.5% chance of you getting the Full House.
Lesson 2: What are my chances of getting a full house on the river?
If it didn’t happen on the turn, your chances usually don’t change all too much, but let’s check. You’ve still got 4 outs and now 46 unseen cards left. 4/46 is about .087 or around an 8.7% chance of hitting it on the river. A .2% difference.
Lesson 3: How about the chances of getting the boat on the turn OR the river?
Like the previous examples, to figure your chance of something happening on multiple events, you need to calculate the chance of it NOT happening first. On the turn it won’t happen 43/47 times. On the river it won’t happen 42/46 times. 43/47 is .915, and 42/46 is .913. Multiply them and get .835, or 83.5% chance of it not happening. Invert that and you get a 16.5% of getting at least a full house by the showdown.
Lesson 4: What do you mean by at least a Full House?
Since we figured the chances to NOT get dealt a full house, the chances are built in if the turn and river are two Kings, two Nines or a King and a Nine. If you are dealt two cards both of either King or Nine, it’ll be four of a kind and not a King and Nine 33% of the time. Think of it as being dealt one card then the other. What are the chances of the first card matching the second? Whether it’s a King or Nine, there will be only one unaccounted for, but two of the other. That’s 1/3, or 33%.
Lesson 5: Then what are my chances of getting four of a kind?
This one requires a little more thought. It doesn’t matter which card we’re hoping for. We need to first get a full house on the turn. According to lesson #1, the chance of that happening is .085. The chance of getting the same card we got on the turn is 1/46. There’s only one out, and the usual 46 unseen cards. 1/46 is around .022 or one fifth of a percent. It will be Kings half of the time and Nines the other half.
Is this making any sense? If you really want to be a master of calculating odds, you need to see these calculations in action, over and over. Like anything else, practice makes perfect. In online games, especially with very few if any tells (shown cards), statistical knowledge becomes the main factor when choosing whether to bet, call or fold.
If you do have a hand that you know can’t lose – you have the nut. Bet like crazy.
While there is a lot more to Texas Hold’em poker than this, this should open your eyes to more things about the game of poker than just the cards and their statistics.
Yes, you DO need to know your general chances of pulling what types of hands, but if you learn to study your opponents, they will tell you their hands and you’ll be able to beat them without even knowing yours.
RETURN ON INVESTMENT (ROI)
When the stakes required to play a game of Texas Hold’em increase, there is not a proportional increase in the average winnings or money flow because most players, especially at the start of play, play tighter at higher stakes.
Here’s how that works.
Higher stakes cause players to be more cautious. Pots do not grow proportionately as the stakes and blinds increase. Your return on investment will therefore decrease as the minimum blind goes up.
Most major online casinos release data on hands played (for a price) on a regular basis. A recent study (June 2004) from one of the largest online casinos, based on several million actual hands of poker played, revealed that the return on investment varies quite a bit based on the maximum bet.
In the $2 games, the value of the winning pot varied from 28 to 37 time the Big Blind (BB) – the most you would have to invest to see the flop (short of raises). The average pots were in the $60 range. With the right cards, you could expect a return of 3000% on a winning hand.
As you can see in the chart above, this ratio falls as the Blinds go up. In the $200 game, with pots averaging $600 1200, the ratio averages 5.5:1. Sure, greater overall winnings, but much greater risk based on the investment you have to make to see the flop. Also notice the volatility or variance of this ratio. On the high stakes tables, play is very tight and often passive, so the ratio remains very narrow – pots are predictably 5 6 times the Big Blind. At the smaller stakes tables, there is considerably more volatility, indicative of a lot looser players and more aggressive playing styles.
The $1/$2 tables are the loosest with pots ranging from 28 37 times the Big Blind.
Are Low Stake Tables Faster?
Not necessarily. Texas Hold’em is the king of fast play. Several $1000 plus pots were played in less than a minute and ranged as long as 6 minutes – the same range for the small stake tables. Over all, the average length of an internet Poker game today is just over one minute or 50-60 hands per hour.
In higher stakes games, one thing is quite clear. There are a higher percentage of tighter and aggressive players at these tables than at the small stake games. That means there are more sharks at the big tables and a much better chance that you will be one of the fish. The smart thing to do here is to stay away from these kinds of tables.
Given the fact that the return on investment is lower at the high stake games, that the average level of play is much more aggressive ant that a much larger stake is required, there is very little opportunity to be a consistent winner on tables with $50 and up blinds.
“All of the recent research points to $5/$10 Limit tables as ideal combination of risk and reward.” Insider Tip
FACT! When the average loose gambler loses, he or she keeps on playing in an attempt to recover the loss. This is irrational and unplanned play and can be very expensive.
On the other hand, when most innate gamblers win, they forget all about their losses and conclude incorrectly that they have finally learned how to win or that their luck has finally changed. They express what is an irrational optimism at this point – a totally unfounded and undeserved optimism that keeps them in the game until they revert back to a losing streak.
Sharks exploit this irrational playing style in gamblers to generate a continuous income.
OUTS ODDS CHART
THE RULE OF FOUR-TWO
The rule of four-two is an easier way to figure the odds for any situation where you know your outs. It is not completely accurate but it will give you a quick ballpark figure of your chances for making a hand. Here is how it works.
With two cards to come after the flop you multiply your number of outs by four. With one card to come after the turn, you multiply you number of outs by two. This will give you a quick figure to work with. If you have a four card flush after the flop you have nine outs. With two cards to come, you multiply the nine by four and you get 36 percent chance of making the flush.
The chart shows the true odds at 35 percent. With one card to come you multiply nine by two and get 18 percent. The chart shows that the true figure is 19.6. It is not completely accurate but it is pretty close, and it is an easy calculation to do in your head.
How to calculate hand odds (the longer way):
Once you know how to correctly count the number of outs you have on a hand, you can use that to calculate what percent of the time you will hit your hand by the river. Probability can be calculated easily for a single event, like the flipping of the river card from the turn. This would simply be: Total Outs / Remaining Cards. For two cards however, like from the flop to the river, it’s a bit more complicated. This is calculated by figuring the probability of your cards not hitting twice in a row. This can be calculated as shown below:
Flop to River % = 1– [((47 – Outs) / 47) * ((46 – Outs) / 46) ]
Turn to River % = (47 – Outs) / 46
The number 47 represents the remaining cards left in the deck after the flop (52 total cards, minus 2 in our hand and 3 on the flop = 47 remaining cards). Even though there might not technically be 47 cards remaining, we do calculations assuming we are the only players in the game. To illustrate, here is a two overcard draw, which has 3 outs for each overcard, giving a total of 6 outs for a top pair draw:
Two overcard draw = 1 – [ (47 – 6)/47 * (46 – 6)/46]
= 1 – [ (41/47) * (40/46) ]
= 1 – [0.87 * 0.87]
= 1 – 0.76
= 0.24
= 24% Chance to Draw Overcards from Flop to River
However, most of the time we want to see this in hand odds, which will be explained after you read about pot odds. To change a percent to odds, the formula is:
Odds = (1/Percentage) – 1
Thus, to change the 24% draw into an odd we can use, we do the following:
Odds = (1/24% Two Overcard Draw ) – 1
= (1/.24) – 1
= 4.17 – 1
= 3.17 or approx 3.2
ANALYZING PROBABILITIES IN DEPTH
You many want to skip this section and go back to it later. Some of this is pretty deep. But important!
Getting a handle on the probability of being dealt various poker hands is one of the most important valuable skills a player can have. We present a number of different ways to do these calculations, from a rough guesstimate system called the 2 4 Rule to the actual combination math.
The first odds calculation that must be made is to determine the total number of possible poker hands in a deck.
As we’ve shown, a poker hand consists of 5 cards drawn from a deck of 52 cards. Therefore, the number of combinations is
COMBIN(52, 5) = 2,598,960
If you use Microsoft Excel, you can duplicate these calculations using the COMBIN factor. COMBIN returns the number of combinations for a given number of items. To find the COMBIN factor in Excel go to INSERT…FUNCTION…MATH & TRIG.
For each of the above “Number of Combinations” we divide by this number to get the probability of being dealt any particular hand.
For the calculations, we will first split out the No Pair hands which include Royal Straight Flushes, Straight Flushes, Flushes, Straights and Nothings. Then we will look at all combinations that have at least 1 pair. The cards in a hand without any pairs will have 5 different denominations selected randomly from the 13 available (2, 3, 4…Ace). Also, each of the 5 denominations will select 1 suit from the four available suits. Thus the total number of no pair hands will equal:
COMBIN(13,5) * (COMBIN(4,1))^5 = 1287 * 1024 = 1,317,888
A Straight Flush is made up of 5 consecutive cards in the same suit and may have a high card of 5, 6, 7, 8, 9, 10, Jack, Queen, King or Ace for a total of 10 different ranks. Each of these may be in any of 4 suits. Thus, there are 40 possible Straight Flushes. An Ace high Straight Flush is a Royal Flush. Since there are only 4 different suits there are only 4 possible Royal Straight Flushes. When we subtract the 4 Royal Straight Flushes from the total of 40 Straight Flushes we are left with 36 other Straight Flushes that are King high or less.
A Flush consists of any 5 of the 13 cards from a particular suit. There are 4 possible suits. The number of possible Flushes is:
COMBIN(13,5) * 4 = 5,148. However, this includes the 40 possible Straight Flushes. When we subtract these out, we are left with:
5,148 – 40 = 5,108 possible ordinary flushes.
A straight consists of 5 cards with consecutive denominations and may have a high card of 5, 6, 7, 8, 9, 10, Jack, Queen, King or Ace for a total of 10 different ranks. Each of these 5 cards may be in any of the 4 suits. Thus, there are 10 * 4^5 = 10,240 different possible straights. However, this total includes the 40 possible Straight Flushes. Thus, we subtract 40, which leaves us with 10,200 possible ordinary straights.
Finally, we come to the nothing hands which are basically all the left over garbage. This is simply the total number of No Pair hands minus all the good stuff. This gives us: 1,317,88 4 36 5,108 10,200 = 1,302,540 nothing hands.
How about the odds of getting 1 pair or better?
A hand with just 1 pair has 4 different denominations selected randomly from the 13 available denominations. 3 of these denominations will select 1 card randomly from the 4 available suits. The 4th denomination will select 2 cards from the available 4 suits. Finally, the pair can be any one of the four available denominations. Thus, the calculation is: COMBIN(13, 4) * (COMBIN(4, 1))^3 * COMBIN(4, 2) * 4 = 1,098,240 possible hands that have just one pair.
The calculation for a hand with two pairs is similar. We will have 3 random denominations taken from the 13 available. Two of these denominations will use 2 of the four available suits while the third denomination selects 1 of the four available suits. The singleton card may be any one of the three denominations. Thus, the calculation becomes: COMBIN(13, 3) * (COMBIN(4, 2))^2 * COMBIN(4, 1) * 3 = 123,552 possible hands with 2 pairs.
Three of a kind is calculated in a similar manner. There will be 3 different denominations from the 13 possible denominations. One denomination will select 3 of the 4 available suits while the other two denominations select 1 card from each of the 4 possible suits. Finally, the three of a kind can be in any of the three denominations. The calculation becomes: COMBIN(13, 3) * COMBIN(4, 3) * (COMBIN(4, 1))^2 * 3 = 54,912 possible hands with 3 of a kind.
The next calculation will be for a Full House. A Full House only uses 2 of the 13 denominations. One of these will select 3 cards from the 4 available while the other selects 2 cards from the 4 available. Finally, the denomination that has 3 cards can be either one of the 2 denominations that we are using. This gives us:
COMBIN(13, 2) * COMBIN(4, 3) * COMBIN(4, 2) * 2 = 3,744 possible Full Houses.
The final calculation is for 4 of a kind. Again, we will select 2 denominations from 13 available. One of these will select 4 cards from the 4 available (obviously the only way to do this is to take all four cards) while the other denomination takes 1 of the available 4 cards. The denomination that has 4 of a kind can be either one of the 2 available denominations.
Thus, the calculation becomes: COMBIN(13, 2) * COMBIN(4, 4) * COMBIN(4, 1) * 2 = 624 different ways of being dealt a 4 or a kind.
Poker Odds From The Turn
Many players who really understand Hold’em odds still tend to forget that the ‘turn’ can change their odds dramatically. It’s true that for a flush draw, the card odds are 1.9 to 1 from the flop to the river. However, this is a theoretical situation where it assumes there is no additional betting on the turn. Typically this is not going to be the case so you will need to recalculate your card and pot odds.
We will use the flush calculation example again and run through it 100 times assuming there was $20 in the pot on the flop with two $5 bets. On the turn, this leaves $30 in the pot, plus a $10 bet from your opponent to call.
Cost to Play = 100 hands * $10 to call on turn = $1,000
Pot Value = $30 + $10 bet + $10 call
Odds to Win = 4.1:1 or 19% (from the turn)
Total Hands Won = 100 * Odds to Win (19%) = 19 wins
Net Profit = Net Cost to Play + (Total Times Won * Pot Value)
= $1,000 + (19 * $50)
= $1,000 + $900
= $100 Profit
Now, you can see that what was a very profitable draw on the flop suddenly turned into a not so great draw on the turn. This is because by not hitting your flush by the turn, it lowered your chances of making a flush by the river. The odds thus increased to 4.1 to 1 instead of 1.9 to 1. So even though the pot odds remained the same at 4:1, because the card odds went down, this flush draw has now become unprofitable.
Realizing the dynamic changes in your odds is extremely important so that you don’t go making incorrect draws based on odds from the flop. Just remember that your odds essentially double from the flop to the turn, so adjust your play accordingly.
Each entry in the following table is the result of 1,000,000 simulated hands of Texas Hold’em played to the showdown and represents the percentage of pots won (including partial pots in the case of splits) by the indicated hand against the indicated number of opponents holding random hands.
The study shows a very clear correlation between your odds of success against the number of players. Notice the JJ, TT, 99 anomaly where the power of these cards increase dramatically over perceived better pocket cards depending on how many players are left.
Hand
How to work this chart…
The numbers below represent number of opponents..
The 2 Hole Cards are represented on the left of the numbers.
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This table represents the percent chance of any particular holdem hand to win if it is played to the river. One will notice the importance of having good cards and the difference of being suited. Also notice that any hand that is suited does better than its unsuited version.
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